We propose a theoretical framework that generalizes simple and fast algorithms for hierarchical agglomerative clustering to weighted graphs with both attractive and repulsive interactions between the nodes. This framework defines GASP, a Generalized Algorithm for Signed graph Partitioning11Code available at: https://github.com/abailoni/GASP, and allows us to explore many combinations of different linkage criteria and cannotlink constraints. We prove the equivalence of existing clustering methods to some of those combinations and introduce new algorithms for combinations that have not been studied before. We study both theoretical and empirical properties of these combinations and prove that some of these define an ultrametric on the graph. We conduct a systematic comparison of various instantiations of GASP on a large variety of both synthetic and existing signed clustering problems, in terms of accuracy but also efficiency and robustness to noise. Lastly, we show that some of the algorithms included in our framework, when combined with the predictions from a CNN model, result in a simple bottom-up instance segmentation pipeline. Going all the way from pixels to final segments with a simple procedure, we achieve state-of-the-art accuracy on the CREMI 2016 EM segmentation benchmark without requiring domain-specific superpixels.

Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments, or equivalently to detect closed contours. Most prior work either requires seeds, one per segment; or a threshold; or formulates the task as multicut / correlation clustering, an NP-hard problem. Here, we propose an efficient algorithm for graph partitioning, the "Mutex Watershed". Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need for explicit seeds or a tunable threshold. We also prove that this simple algorithm solves to global optimality an objective function that is intimately related to the multicut / correlation clustering integer linear programming formulation. The algorithm is deterministic, very simple to implement, and has empirically linearithmic complexity. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the Mutex Watershed gives the best results currently known for the competitive ISBI 2012 EM segmentation benchmark.